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The work reported in this paper deals with the study of a coupled system for fractional terminal value problems involving ψ-Hilfer fractional derivative. The existence and uniqueness theorems to the problem at hand are investigated. Besides, the stability analysis in the Ulam–Hyers sense of a given system is studied. Our discussion is based upon known fixed point theorems of Banach and Krasnoselskii. Examples are also provided to demonstrate the applicability of our results. [ABSTRACT FROM AUTHOR]

To have a more realistic model, in this paper, This manuscript is devoted to investigating a fractional-order mathematical model of Kouidere et al. That describes the dynamics of spread of African swine fever virus (ASFV). The aim of this work is to protect susceptible pigs from the virus, In our model, by including three controls which represent: the iron fencing and spraying pesticides and get rid. The aims of this paper is to reduce the number of infected pigs and ticks by using optimal control strategy and fractinal order derivation. Pontryagin's maximal principle is used to describe optimal controls with Caputo time-fractional derivative and the optimal system is resolved in an iterative manner. Numerical simulations are presented based on the presented method. We finished tis article with a conclusion. [ABSTRACT FROM AUTHOR]

A characteristic of ecosystems is the existence of manifold of independencies which are highly complex. Various mathematical models have made considerable contributions in gaining a better understanding of the predator–prey interactions. The main components of any predator–prey models are, firstly, how the different population classes grow and secondly, how the prey and predator interacts. In this paper, the two populations’ growth rates obey the logistic law and the carrying capacity of the predator depends on the available number of prey are considered. Our aim is to clarify the relationship between models and Holling types functional and numerical responses in order to gain insights into predator interferences and to answer an important question how competition is carried out. We consider a predator–prey model and a two-predator one-prey model to explain the idea. The novel approach is explained for the mechanism measurement of predator interference through depending on numerical response. Our approach gives good correspondence between an important real data and computer simulations. [ABSTRACT FROM AUTHOR]

The GM (1, N) model is an important prediction model among gray models, but the conventional GM (1, N) model shows big errors sometimes mainly due to the structural inconsistency between the gray differential equation for parameter estimation and the whitening equation for prediction. The paper derives the modified model of conventional gray differential equation based on the whitening equation, and the gray differential equation obtained in this way has the structure basically consistent with the structure of whitening equation. The paper estimates parameters with the gray differential equation of modified GM (1, N) model and gives three parameter estimation methods with different background values. The time response equation for conventional GM (1, N) model generally has no analytical solution, but the paper gives a specific solving method for the time response equation of modified model. The example proves that the modified GM (1, N) model given and the models built with the methods proposed improve the simulation and prediction accuracy significantly compared with the conventional model. [ABSTRACT FROM AUTHOR]

In the gray prediction, the GM(1,1) model has been applied widely, but the conventional GM(1,1) model prediction shows big errors sometimes. Many researchers improve prediction accuracy by changing the basic structure of model. The paper tries to make improvements in the models parameter estimation on the basis of new background value optimization without changing the models structure, and gives the following four methods: (1) estimating the parameter of gray model GM(1,1) with the optimized value of exponential curve as the background value; (2) estimating the parameter of gray model GM(1,1) with the optimized value of power function curve as the background value; (3) estimating the parameter of gray model GM(1,1) with the optimized value of polynomial curve as the background value; (4) estimating the parameter of gray model GM(1,1) with the optimized value of interpolation function as the background value. The last part of the paper builds the GM(1,1) model of China's total clean energy consumption with the four improved methods. The simulation and prediction results show that these methods all improve the prediction accuracy of model significantly. [ABSTRACT FROM AUTHOR]

This paper is concerned with the growth-driven shape-programming problem, which involves determining a growth tensor that can produce a deformation on a hyperelastic body reaching a given target shape. We consider the two cases of globally compatible growth, where the growth tensor is a deformation gradient over the undeformed domain, and the incompatible one, which discards such hypothesis. We formulate the problem within the framework of optimal control theory in hyperelasticity. The Hausdorff distance is used to quantify dissimilarities between shapes; the complexity of the actuation is incorporated in the cost functional as well. Boundary conditions and external loads are allowed in the state law, thus extending previous works where the stress-free hypothesis turns out to be essential. A rigorous mathematical analysis is then carried out to prove the well-posedness of the problem. The numerical approximation is performed using gradient-based optimisation algorithms. Our main goal in this part is to show the possibility to apply inverse techniques for the numerical approximation of this problem, which allows us to address more generic situations than those covered by analytical approaches. Several numerical experiments for beam-like and shell-type geometries illustrate the performance of the proposed numerical scheme. [ABSTRACT FROM AUTHOR]

Mathematical models have shed light on the dynamics of HIV- 1 infection in vivo. In this paper, we generalize continuous mathematical models of drug therapy for HIV-1 by Perelson et al. (Science 271:1582–1586, 1996) and Perelson and Nelson (SIAM Rev 41:3–44, 1999) on time scales, i.e., a nonempty closed subset of real numbers in order to derive new discrete models that predict the total concentration of plasma virus as a function of time. One of our main goals is to compare discrete mathematical models with the continuous model in Perelson et al. (1996) where HIV infected patients were given protease inhibitors and sampled frequently thereafter. For the comparison, we use experimental data collected in Perelson et al. (1996) and estimate the parameters such as the virion clearance rate and the rate of loss of infected cells by fitting the total concentration of plasma virus to this data set. Our results show that discrete systems describe the best fit. In the previous models of this study, the efficacy of protease inhibitor is assumed to be perfect. Motivated by Perelson and Nelson (1999), we end the paper with a mathematical model of imperfect protease inhibitor and reverse transcriptase (RT) inhibitor combination therapy of HIV-1 infection on time scales with its stability analysis. [ABSTRACT FROM AUTHOR]

In many industrial applications, reactive filters, which remove toxic substances using chemical reactions, prove very efficient and versatile. One such example is the removal of sulphur dioxide from flue gas by converting it into liquid sulphuric acid. In this paper, we extend a model developed in Kiradjiev et al. (SIAM J Appl Math 82:591–619, 2021) for the operation of a reactive filter to consider the long-time behaviour and determine how the filter becomes clogged with liquid and becomes inefficient. We explore how the parameters that govern the operation of the filter affect its efficiency and discuss optimal filtration strategies. We also consider a distinguished asymptotic limit in terms of the key dimensionless parameters and obtain an explicit solution that compares well with the numerical solution to the full problem. Finally, we compare the results of our model with those from a simplified model for a reactive filter considered in Breward and Kiradjiev (J Eng Math, 129(14):1–28, 2021). [ABSTRACT FROM AUTHOR]

A long series of previous papers have been devoted to the (one-dimensional) moment problem with nonnegative rational measure. The rationality assumption is a complexity constraint motivated by applications where a parameterization of the solution set in terms of a bounded finite number of parameters is required. In this paper we provide a complete solution of the multidimensional moment problem with a complexity constraint also allowing for solutions that require a singular measure added to the rational, absolutely continuous one. Such solutions occur on the boundary of a certain convex cone of solutions. In this paper we provide complete parameterizations of all such solutions. We also provide errata for a previous paper in this journal coauthored by one of the authors of the present paper. [ABSTRACT FROM AUTHOR]

In the paper we propose a model describing interaction characteristics of hydrated ions in the mixtures of uni-univalent electrolytes with a common ion. The main factor taken into consideration is a variation of the hydrate shell size in the mixture from its value in the binary solution. Account of this factor gives a possibility to calculate the corresponding variation of the potential describing the mutual ions interaction. The potential is the sum of the electrostatic potential of ions and potential resulted from deformation of approaching hydrated shells. On this basis it is possible to explain the behavior of curves describing the activity coefficients in the mixtures of electrolytes in the range of concentrations from zero to several moles per liter. The results obtained by a numerical modelling are verified by comparing with the available experimental data. [ABSTRACT FROM AUTHOR]

Delayed models are a better representation of the nature of HIV. In the present paper, a multi-delayed model of HIV with combination drug therapy has been analysed. Effect of the immune response in the form of effector cell response has also been included to make the model more justified. The threshold properties related with the basic reproduction number R0 have been discussed. The local and global properties of the model have been analysed. Extensive numerical simulations have been performed to show the impact of highly effective drug on the concentration of virus. The numerical simulations and the result proved has led to the conclusion that a highly effective drug when combined with a less effective drug, can very efficiently bring down the viral load to undetectable levels. [ABSTRACT FROM AUTHOR]

In this paper, we deal with Caputo-type fractional differential inequality where there is a low-order fractional derivative with the term polynomial source. We investigate the nonexistence of nontrivial global solutions in a suitable space via the test function technique and some properties of fractional integrals. Finally, we demonstrate three examples to illustrate our results. The presented results are more general than those in the literature, which can be obtained as particular cases. [ABSTRACT FROM AUTHOR]

Fall armyworm (Spodoptera frugiperda), a highly destructive and fast spreading agricultural pest native to North and South America, poses a real threat to global food security. In this paper, to explore the dynamics and implications of fall armyworm outbreak in a field of maize biomass, we propose a new dynamical system for maize biomass and fall armyworm interaction via Caputo fractional-order operator, which is not only a nonlocal operator but also contains all characteristics concerned with memory of the dynamical system. We define the basic reproduction number, which represents the average number of newborns produced by one individual female moth during its life span. We establish that the basic reproduction number is a threshold quantity, which determines persistence and extinction of the pest. Finally, we simulate the Caputo system using the Adam–Bashforth–Moulton method to illustrate the main results. [ABSTRACT FROM AUTHOR]

This paper presents a state-of-art survey of the research on fractional-order (FO) modeling with parameter identification, and FO estimation methods for state of charge (SOC), state of health (SOH), and remaining usage life (RUL) of lithium-ion batteries (LIBs) mainly in recent five years. FO electrochemical models and six different types of FO equivalent circuit models (ECMs) are introduced in detail. Then, the corresponding tuning algorithm for parameters of these FO models are also provided in brief. Moreover, FO estimation methods for SOC are listed and analyzed, mainly including FO observers, and FO Kalman filters (FO-KFs). SOH and RUL estimation is another vital aspect for LIBs ageing and degradation monitoring, thus FO estimation methods proposed in recent research within five years are all listed. Finally, some suggestions that may be helpful for further research are proposed in conclusion. [ABSTRACT FROM AUTHOR]

Every year, millions of people worldwide (including a major portion in China) are suffering from lung cancer disease (Chinese report of Smoking and Health 2017). The aim of this paper is to develop a new fuzzy soft expert system which can be used to predict lung cancer disease. A prediction process using this fuzzy soft expert system is composed of four main steps: (1) Transform real-valued inputs into fuzzy numbers. (2) Transform fuzzy numbers of data into fuzzy soft sets. (3) Reduce, using normal parameter reduction method, the obtained family of fuzzy soft sets into a new family of fuzzy soft sets. (4) Use the proposed algorithm to get the output data. An experiment is conducted on forty five patients (thirty males, fifteen females, all are cigarette smokers) who endure treatment in the Respiratory Department of Nanjing Chest Hospital, China. The number of training data taken was 55 records, and the remaining 45 records were used for the testing process in our system by using weight loss, shortness of breath, chest pain, persistence a cough, blood in sputum, and age of patients. The quantized accuracies of the proposed system were found to be 100 % . In this work, we developed a fuzzy soft expert system based on fuzzy soft sets; we used a fuzzy membership functions and an algorithm to predict those patients who may suffer lung cancer. In this way, it is possible to conclude that the use of fuzzy soft expert system can produce valuable results for lung cancer detection. It is found that the fuzzy soft expert system developed is useful to the expert doctor to decide if a patient has lung cancer or not. Finally, we introduce comparison diagnosed between our proposed system and the fuzzy inference system. [ABSTRACT FROM AUTHOR]

In this paper, we establish the existence of periodic solution of a delayed predator-prey model with M -predators and N -preys over the time scales. We derive sufficient conditions for the existence of a periodic solution with the help of continuation theorem of coincidence degree theory. At the end, we give an example with numerical simulations to illustrate our analytical findings. [ABSTRACT FROM AUTHOR]

We investigate a well-known phenomenon of variational approaches in image processing, where typically the best image quality is achieved when the gradient flow process is stopped before converging to a stationary point. This paradox originates from a tradeoff between optimization and modeling errors of the underlying variational model and holds true even if deep learning methods are used to learn highly expressive regularizers from data. In this paper, we take advantage of this paradox and introduce an optimal stopping time into the gradient flow process, which in turn is learned from data by means of an optimal control approach. After a time discretization, we obtain variational networks, which can be interpreted as a particular type of recurrent neural networks. The learned variational networks achieve competitive results for image denoising and image deblurring on a standard benchmark data set. One of the key theoretical results is the development of first- and second-order conditions to verify optimal stopping time. A nonlinear spectral analysis of the gradient of the learned regularizer gives enlightening insights into the different regularization properties. [ABSTRACT FROM AUTHOR]

As the world is moving into the 4th Industrial revolution, commonly referred to as Industry 4.0, mass production is giving way to mass customization. This shift necessitates a change in the design of the assembly line with flexibility and multi variant manufacturing taking precedence. These kind of assembly lines that can manufacture multiple models without a predetermined manufacturing time are referred to as mixed model stochastic assembly lines. This paper looks at how such a mixed model stochastic assembly line can be modelled. This is done by taking the case study of a water bottling plant that is being developed at the university where this research is conducted. The water bottling plant needs to produce 500 ml and 750 ml bottles of clean drinking water which can be used within the university or be sold externally. The primary aim of this paper is to develop a MATLAB model for the water bottling plant and study the results from the simulations. These results will shed light on how the model can be optimized thereby contribute to the niche area of Mixed Model Stochastic Assembly Line Balancing. [ABSTRACT FROM AUTHOR]

This mathematical contribution is addressed towards the wide interface of life and human sciences that exists between biological and environmental information. Like very few other disciplines only, the modeling and prediction of genetical data is requesting mathematics nowadays to deeply understand its foundations. This need is even forced by the rapid changes in a world of globalization. Such a study has to include aspects of stability and tractability; the still existing limitations of modern technology in terms of measurement errors and uncertainty have to be taken into account. In this paper, the important role played by the environment is rigorously introduced into the biological context and connected with employing the theories of optimization and dynamical systems. Especially, a matrix-vector and interval concept and algebra are used; some special attention is paid to splines. From data got by DNA microarray experiments and environmental measurements we extract nonlinear ordinary differential equations. This is done by Chebychev approximation and semi-infinite optimization. Then, time-discretized dynamical systems are studied. By a combinatorial algorithm which constructs and follows polyhedra sequences, the region of parametric stability is detected. This is used for testing and maybe improving the goodness of the achieved model. We analyze the topological landscape of gene-environment networks in terms of structural stability which we characterize. This pioneering practically motivated and theoretically elaborated work is devoted to a contribution to better health care, progress in medicine, better education, and to recommending more healthy living conditions. The present paper mainly bases on the authors’ and their coauthors’ contributions of the last few years, it critically discusses structural frontiers and future challenges, while respecting related research contributions, giving access and referring to alternative concepts that exist in the literature. [ABSTRACT FROM AUTHOR]

Until finding medicines or vaccines for newly outbreaking infectious diseases, the mostly applying control strategies are maintaining social distancing, quarantining suspected exposures, and isolating infectious people. This paper proposes a mathematical model for control strategies of COVID-19, considering population sub-classes susceptible (S), Exposed (E), Quarantined (Q), Infected (I), Isolated (J), and Recovered (R). Mathematical models consisting of these sub-classes are called SEQIJR type models. The basic reproduction number of the proposed mathematical model is derived based on the next-generation matrix. Based on the basic reproduction number of the model, the disease dies out and the disease persists parameter spaces are determined. The effects of the rates and applying stages of the control strategies on the spread of the disease are explained. Outcomes of the proposed model are helpful to identify suitable stages and strengths of the control strategies to control the disease, considering available health care capacity. [ABSTRACT FROM AUTHOR]

How will bounded rationality influence telecommunication network fluctuations? Recently, there has been an increased research interest in telecommunication network pricing, which leads to many proposals for new pricing schemes motivated by different objectives namely: to maximize service provider's revenue, to guarantee fairness among users and to satisfy quality of service (QoS) requirements for differentiated network services. In the present paper, we consider a system with N rational service providers (SPs) that offer homogeneous telecommunication services to bounded rational costumers. All SPs offer the same services and seek to persuade more customers in the same system, we model this conflict as a noncooperative game. On the one hand, each SP decide his policies of price and QoS in order to maximize his profit. One the other hand, we assume that the customers are boundedly rational and make their subscription decisions probabilistically, according to Luce choice probabilities. Furthermore, the customers decide to which SP to subscribe, each one may migrate to another SP or alternatively switch to "no subscription state" depending on the observed price/QoS. In this work, we have proved through a detailed analysis the existence and uniqueness of Nash equilibrium. We evaluate the impact of user's bounded rationality on the equilibrium of game. Using the price of anarchy, we examine the performance and efficiency of equilibrium. We have shown that the SPs have an interest in confusing customers, which means more than the customers are irrational, the SPs earn more. [ABSTRACT FROM AUTHOR]

The focus of this paper is to develop a new partial differential equation model for the pattern formation of the human pathogen Staphylococcus aureus, starting from a newly developed model of selected gene regulation mechanisms. In our model, we do not only account for the bacteria densities and nutrient concentrations, but also for the quorum sensing and biofilm components, since they enable bacteria to coordinate their behavior and provide the environment in which the colony grows. To this end, we model the relevant gene regulation systems using ordinary differential equations and therefrom derive our evolution equations for quorum sensing and biofilm environment by time-scale arguments. Furthermore, we compare and validate our model and the corresponding simulation results with biological real data observations of Staphylococcus aureus mutant colony growth in the laboratory. We show that we are able to adequately display the qualitative biological features of pattern formation in selected mutants, using the parameter changes indicated by the gene regulation mechanisms. [ABSTRACT FROM AUTHOR]

The paper deals with monocular image-based sense and avoid assuming constant aircraft velocities and straight flight paths. From very limited two dimensional image information it finally characterizes the whole three dimensional collision situation by estimating the time to closest point of approach, the horizontal relative distance and its direction and the vertical relative distance also. The distances are relative to the intruder aircraft horizontal and vertical sizes. The overall estimated relative distance is the closest between the two aircraft in three dimension. So finally, every important information can be extracted to be used in a collision decision. The applicability of the developed method is presented in software-in-the-loop simulation test runs. Several intruder size and speed values are considered together with trajectories covering the whole three dimensional space. The horizontal intruder flight directions relative to the own aircraft cover 360∘ and the intruder can come from below ar above also. Detailed evaluation and discussion of the results is also included. Finally, the missed detection rate results to be superior (below 3% in every test scenario) though the false alarm rate results a bit high between 7-14%. [ABSTRACT FROM AUTHOR]

In this paper, structure-preserving model reduction methods for second-order systems are investigated. By introducing an appropriate parameter, the second-order system is represented by a strictly dissipative realization and the H2 norm of the strictly dissipative system is discussed. Then, based on the Krylov subspace techniques, two model reduction methods are proposed to reduce the order of the strictly dissipative system. Further, the reduced second-order systems are obtained. Moreover, according to the factorization of the error system, the H2 error bounds are represented by the Kronecker product and the vectorization operator. Finally, two numerical examples illustrate the efficiency of our methods. [ABSTRACT FROM AUTHOR]

Understanding the relation between large-scale and small-scale functionality is an acute problem in large systems. This paper presents possible structures that capture those relations. Here “structure” means a functor on a category, in this case the category of “combinatorial systems” that model manufacturing systems and supply networks. The details of the dynamics of these systems are frequently the domain of specialists and are ill understood outside such areas. Clustering activities in subsystems introduces covers as scales of analysis. This allows the development of sheaves of schedules and their properties which are analyzed by an adaptation of Čech cohomology. These are applied in supply networks to characterize the fragmentation of these networks and to characterize the conditions for local changes in schedules to extend to the whole system. The intent of this paper is to see sheaves and their cohomology as analytical probes that can enhance our understanding of large-scale systems. [ABSTRACT FROM AUTHOR]

In this paper, a new reliable algorithm called the multistage homotopy analysis method (MHAM) based on an adaptation of the standard homotopy analysis method (HAM) is presented to solve a time-fractional enzyme kinetics. This enzyme–substrate reaction is formed by a system of nonlinear ordinary differential equations of fractional order. The new algorithm is only a simple modification of the HAM, in which it is treated as an algorithm in a sequence of small intervals (i.e. time step) for finding accurate approximate solutions to the corresponding systems. Numerical comparisons between the MHAM and the classical fourth-order Runge–Kutta method in the case of integer-order derivatives reveal that the new technique is a promising tool for nonlinear systems of integer and fractional order. [ABSTRACT FROM AUTHOR]

Our aim in this work is to synthesize optimal feeding strategies that maximize, over a time period, the biogas production in a continuously filled bioreactor controlled by its dilution rate. Such an anaerobic process is described by a four-dimensional dynamical system. Instead of modeling the optimization of the biogas production as a Lagrange-type optimal control problem, we propose a slightly different optimal control approach in this paper: We study the minimal time control problem to reach a target point, which is chosen in such a way that it maximizes the biogas production at steady state. Thanks to the Pontryagin maximum principle and the geometric control theory, we provide an optimal feedback control for the minimal time control problem, when the initial conditions are taken within the invariant and attractive manifold of the system. The optimal synthesis exhibits turnpike and anti-turnpike singular arcs and a cut locus. [ABSTRACT FROM AUTHOR]

Nonlocal interactions, which have attracted attention in various fields, result from the integration of microscopic information such as a transition possibility, molecular events, and signaling networks of living creatures. Nonlocal interactions are useful to reproduce various patterns corresponding to such detailed microscopic information. However, the approach is inconvenient for observing the specific mechanisms behind the target phenomena because of the compression of the information. Therefore, we previously proposed a method capable of approximating any nonlocal interactions by a reaction-diffusion system with auxiliary factors (Ninomiya et al., J Math Biol 75:1203-1233, 2017). In this paper, we provide an explicit method for determining the parameters of the reaction-diffusion system for the given kernel shape by using Jacobi polynomials under appropriate assumptions. We additionally introduce a numerical method to specify the parameters of the reaction-diffusion system with the general diffusion coefficients by the Tikhonov regularization. [ABSTRACT FROM AUTHOR]

This paper proposes a differential equation model of the human vision system. Our model has a hierarchical structure in the known retinal cell neural network, and hence, enable to explain what aspects of behaviors of the vision system is affected by which parameters. As the result, our model possesses the following two characteristics: First, it enable the derivation of an integral equation model with a Mexican-hat shape kernel as a necessary result of mechanisms (reduction of the retinal image resolution and a self-control mechanism formed by non-local interaction), in contrast to previous models that assumed various types of Mexican-hat shapes a priori. Second, it can explain two mutually contradicting phenomena called lightness contrast and assimilation. Moreover, our model explains the reason why lightness optical illusions do or do not occur via the magnitude of a control parameter. [ABSTRACT FROM AUTHOR]

Recent years have seen the introduction of non-local interactions in various fields. A typical example of a non-local interaction is where the convolution kernel incorporates short-range activation and long-range inhibition. This paper presents the relationship between non-local interactions and reaction-diffusion systems in the following sense: (a) the relationship between the instability induced by non-local interaction and diffusion-driven instability; (b) the realization of non-local interactions by reaction-diffusion systems. More precisely, it is shown that the non-local interaction of a Mexican-hat kernel destabilizes the stable homogeneous state and that this instability is related to diffusion-driven instability. Furthermore, a reaction-diffusion system that approximates the non-local interaction system with any even convolution kernel is shown to exist. [ABSTRACT FROM AUTHOR]

Interconnected systems are an important class of mathematical models, as they allow for the construction of complex, hierarchical, multiphysics, and multiscale models by the interconnection of simpler subsystems. Lagrange-Dirac mechanical systems provide a broad category of mathematical models that are closed under interconnection, and in this paper, we develop a framework for the interconnection of discrete Lagrange-Dirac mechanical systems, with a view toward constructing geometric structure-preserving discretizations of interconnected systems. This work builds on previous work on the interconnection of continuous Lagrange-Dirac systems (Jacobs and Yoshimura in J Geom Mech 6(1):67-98, 2014) and discrete Dirac variational integrators (Leok and Ohsawa in Found Comput Math 11(5), 529-562, 2011). We test our results by simulating some of the continuous examples given in Jacobs and Yoshimura (2014). [ABSTRACT FROM AUTHOR]